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- 1. SCIENTIFIC CALCULATORTOPIC : QUADRATIC EQUATIONS (ADDITIONAL MATHEMATICS F4)SUBTOPIC : Discriminant of a Quadratic EquationLearning objective : Understand and use the condition for quadratic equation which is: Two different root Two equal root No rootLearning outcome : Student should be able to state and determine the type of root of quadratic equation.IntroductionIn mathematics, a quadratic equation is a univariate polynomial equation of the second degree. Ageneral quadratic equation can be written in the form ax 2 bx c 0 where x representsa variable or an unknown, and a, b, and c are constants with a ≠ 0.(If a = 0, the equation is a linear equation.)The constants a, b, and c are called respectively, the quadratic coefficient, the linear coefficientand the constant term or free term. Quadratic equations can be solved by : a) factoring (if quadratic equation cannot be factored, then this method will not work) b) completing the square which is : a( x h) 2 k b b2 4ac c) quadratic formulawhich is : x 2aFrom the quadratic formula, b 2 4ac is known as the discriminant. The discriminant givesadditional information on the nature of the roots whether there are any repeated roots. It alsogives information on whether the roots are real or complex, and rational or irrational. Moreformally, it gives information on whether the roots are in the field over which the polynomial isdefined, or are in an extension field, and hence whether the polynomial factors over the field ofcoefficients. This is most easily stated for quadratic and cubic polynomials; for polynomials ofdegree 4 or higher this is more difficult to state. The nature of the roots are given as below:
- 2. Value of the discriminant Type and number of roots Example of graphPositive Discriminant Two Real Rootsb² − 4ac > 0 If the discriminant is a perfect square the roots are rational. Otherwise, they are irrational.Discriminant is Zero One Real Rootb² − 4ac = 0Negative Discriminant No Real Rootsb² − 4ac < 0 Two Imaginary SolutionsAs a conclusion, by using the following module, it will help student to understand more aboutdiscriminant, quadratic equation and develop their own knowledge by using scientific calculator. In order to do this worksheet, student should already know about: Quadratic equation general form ( ax 2 bx c 0 ) Determine the value of a, b, and c Example : x 2 5x 24 0 a=1 b= 5 c = -24
- 3. Factorize quadratic equations Example : x 2 5x 24 0 ( x 3)(x 8) 0 factorize From the result of worksheet, student should found that : a) If b 2 4ac 0 , then the quadratic equation has 2 different roots. b) If b 2 4ac 0 , then the quadratic equation has 2 equal roots. c) If b 2 4ac 0 , then the quadratic equation has no roots.Mapping Form Form 1 Form 2 Form 3 Form 4 Form 5 Form 6Prior knowledgeAlgebraic expressionGraph of functionQuadratic expression & equationsQuadratic equations ( Add math)Activity a) How to use calculator? Use the key MODE to enter the EQN mode Press 3 times and press number 1 Then, press to display the quadratic/ cubic equation screen. Example : Degree ? 2 3
- 4. Use this screen to specify 2 (quadratic) and 3 (cubic) as the degree of equation. Input the value of the coefficients until reach for the final coefficient which is until C for quadratic equation. Example: Use and keys to move between coefficients and make changes Coefficient a? Direction to name 0. view other elements Element value Calculation start and one of the solution will appears. Then, press = keys to view other solution. Use and to scroll all solution. Next, press AC key to returns to the coefficient input screen. Example : Direction to view Variable x1= otherelements name 0. Solution
- 5. EXAMPLE : 6x 2 x 2 0 x1= R I(Degree?) 2 0.25a? 6 = SHIFT R Ib? 1 =c? -2 = x2= R Ix1 0. 5 0.75 ix2 0.7
- 6. b) Worksheet Complete the following table by using scientific calculator. Equation a b c b2 4ac Type of roots x1 x2 x2 2x 3 3x 2 6 x x2 4x 4 x2 2x 1 x2 3 2 x 2 3x 8 c) Worksheet ( Answer)Equation a b c b2 4ac Type of roots x1 x2 Positive discriminantx2 2x 3 1 -2 -3 16 b 2 4ac 0 3 -13x 2 6 x Positive discriminant 3 6 0 36 b 2 4ac 0 0 -2x2 4x 4 Discriminant is zero 1 -4 4 0 b 2 4ac 0 2 2 Discriminant is zerox2 2x 1 1 -2 1 0 b 2 4ac 0 1 1 Negative discriminant x2 3 1 .7 i 1 .7 i 1 0 3 -12 b 2 4ac 02 x 2 3x 8 Negative discriminant 0.75 1.9i 0.75 1.9i 2 3 8 -55 b 2 4ac 0
- 7. d) Conclusion Unequal real roots Equal real roots No real roots b 2 4ac 0 b 2 4ac 0 b 2 4ac 0ExerciseComplete the following table without using scientific calculator. Equation a b c b2 4ac Type of roots Conclusion x2 x 2 20 x 2 36 x 9 x 2 8 x 16 4 x 2 12 x 9 9 x 2 81 25x 2 50
- 8. Exercise AnswerComplete the following table without using scientific calculator. Equation a b c b2 4ac Type of roots Conclusion Positive discriminant x2 x 2 1 1 -2 9 b 2 4ac 0 Unequal 2 Positive discriminant real roots20 x 36 x 9 20 -36 9 576 b 2 4ac 0 Discriminant is zero x 2 8 x 16 1 -8 16 0 b 2 4ac 0 Equal real 2 Discriminant is zero roots 4 x 12 x 9 4 -12 9 0 b 2 4ac 0 9 x 2 81 Negative discriminant -9 0 -81 -2916 b 2 4ac 0 No real Negative discriminant roots 25x 2 50 25 0 -50 -5000 b 2 4ac 0
- 9. ReferencesOoiSooHuat, et al. , Additional Mathematics Form 4, Selangor: NurNiagaSdnBhd, 2005.Zaini B Musa, et al. ,Additional Mathematics Form 4, Selangor: Cerdik Publication SdnBhd,2005.Huraiansukatanpelajaranmatematiktingkatan 1,2dan 3Huraiansukatanpelajaranmatematiktambahantingkatan 4 dan 5Huraiansukatanpelajaranmatematik T tingkatan 6The discriminant in quadratic equation. Retrieved from TheMathWarehousePage:http://www.mathwarehouse.com/quadratic/discriminant-in-quadratic-equation.phpDiscriminant.Retrieved from Wikipedia Page:http://en.wikipedia.org/wiki/Discriminant#Quadratic_formulaQuadratic euqations.Retrieved from Wikipedia Page :http://en.wikipedia.org/wiki/Quadratic_equation

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